Mesh is warped when I add subdivision surface. No amount of loop cuts gets rid of it
After examining your file, I found 2 main problems - one easy to solve, and the other, a bit more time consuming, to say the least.
The first (easy to solve) issue seems to be that you have a good deal of duplicate vertices in your mesh (somewhere in the 400 area). To solve this, select the whole mesh in edit mode (A) and select mesh &gt; clean up &gt; merge by distance from the mesh menu at the top of the viewport.
The second (not so easily solved) issue has to do with the topology of the mesh. Specifically, your mesh has a good number of n-gon's (faces with more than 4 vertices) which is causing the majority of your issues. Blender favors quad-based topology, and though n-gon's or tri's are sometimes unavoidable, they are best kept towards the center of flat faces, otherwise they cause issues with any number of things, from Subdivision Surface Modifiers, to Shading.
The solution, then, is to do your best to re-topologize your mesh to contain primarily quad-based faces (especially near curved edges) . In the image below, I have begun this process by joining many of the vertices to make quad faces. (I did not finish, however, as it is a lot of work for an example), You can see that the areas that are made of quads behave better, and those that have more than 4 vertices do not. I was also able to better define the overall form of the shape by adding the edge loops I left highlighted.

It's better, but it still needs a lot of retopology work to be &quot;perfect&quot;. You will need to spend some time adjusting it manually, I'm afraid. You can see from the image that there are still some artifacts where the the n-gons remain. Check YouTube for guides on topology if you need it.